Question

In an AP sum of first 10 th term is 80 and sum of next 10th term is -280 . Find the AP​

Answer 1

Answer:

The answer will be:

476/19, 404/19, 332/19,...so on.

Step-by-step explanation:

We have given;

S10 = 80;

and, S10-20 (i.e, next 10 term of a10) = -280;

The formula of adding A.P. to n term is given as;

= (n/2)(a + al) [here, al is the last term] or, = (n/2)(a + a +d(n-1))

Hence, for S10, the equation we can write is;

= (n/2)(a + a +d(n-1)) = 80;

= (10/2)(a + a +d(10-1)) = 80;

= 5 (2a + 9d) = 80;

= 2a + 9d = 16; (i)

For the sum of the next 10 terms from a10 or S10-20, the formula will change to:

= (n/2)(a10 + a20) (ii)

( here, a10 will become first term and a20 will be the last term. )

Thus, for equation (ii);

a10 = a + d(10 - 1);

= a + 9d; (iii)

and, a20 = a + d(20 - 1);

= a + 19d; (iv)

Here, the terms will remain 10 because we are using the next 10 terms from a10.

So, n = 10. (for eqaution (ii)) (v)

Now, put the value of equation (iii), (iv), (v) and S10-20, in equation (ii);

= (n/2)(a10 + a20) = -280;

= (10/2)(a +9d + a + 19d) = -280;

= 5 (2a + 28d) = -280;

= 2a + 28d = -56; (vi)

Using substitution method in equation (i) and (ii), we can write;

= (16 - 9d) + 28d = -56;

= 16 + 19d = -56;

= d = (-56 -16) / 19;

= -72/19;

Hence, using the value of d and equation (i), we can derive the value of a.

Thus, = 2a + 9(-72/19) = 16;

= 2a = 16 - 9(-72/19);

= 2a = 16 - (-648/19);

= 2a = 952/19

= a = 476/19;

Therefore, the A.P. will be :

476/19, 404/19, 332/19,...

(I believe you can proceed this A.P. now)

Verify:

We had given;

S10 = 80;

Lets put the value of a and d in the formula of adding A.P.;

= (n/2)(a + a +d(n-1)) = 80;

= (10/2){(476/19) + (476/19) + (-72/19)(10-1)} = 80;

= 5 {(952/19) + (-648/19)} = 80;

= 5 { (952-648)/19} = 80;

= 5 {304 / 19} = 80;

= 5 (16) = 80;

= 80 = 80.

= RHS = LHS

Hence, verified.

We had given;

S10-20 = -280;

Lets put the value of a and d in the formula of adding A.P.;

= (10/2)(a10 + a20) = -280;

= (10/2)(a + d(10-1) + a + d(20-1)) = -280;

= (5)(a + 9d + a + 19d) = -280;

= ((476/19) + 9(-72/19) + (476/19) + 19(-72/19)) = -56;

= ((952/19) + (-648/19) + (-1368/19)) = -56;

= ((952 - 648 - 1368) / 19) = -56;

= (-1064 / 19) = -56;

= - 56 = -56

= RHS = LHS

Hence, verified.

That's all.